3.31.15 \(\int \frac {x (-a+x) (a b-2 b x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)^2} (-b^2+2 b x+(-1+a^2 d) x^2-2 a d x^3+d x^4)} \, dx\) [3015]

3.31.15.1 Optimal result

Integrand size = 71, antiderivative size = 415 \[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 b-2 x+\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (b \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b^2-2 b x+x^2+\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}\right )}{2 d^{5/6}} \]

output

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3 
+x^4)^(1/3)/(2*b-2*x+d^(1/6)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^( 
1/3)))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(-a*b^2*x+(2*a*b+b^2)*x^ 
2+(-a-2*b)*x^3+x^4)^(1/3)/(-2*b+2*x+d^(1/6)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a- 
2*b)*x^3+x^4)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(-a*b^2*x+(2*a*b+b^2)*x^2+(- 
a-2*b)*x^3+x^4)^(1/3)/(b-x))/d^(5/6)+1/2*arctanh((b*d^(1/6)-d^(1/6)*x)*(-a 
*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/3)/(b^2-2*b*x+x^2+d^(1/3)*(-a* 
b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(2/3)))/d^(5/6)
 

3.31.15.2 Mathematica [F]

\[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx \]

input

Integrate[(x*(-a + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)* 
(-b^2 + 2*b*x + (-1 + a^2*d)*x^2 - 2*a*d*x^3 + d*x^4)),x]
 

output

Integrate[(x*(-a + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)* 
(-b^2 + 2*b*x + (-1 + a^2*d)*x^2 - 2*a*d*x^3 + d*x^4)), x]
 

3.31.15.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x (x-a) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (x-a) (x-b)^2} \left (x^2 \left (a^2 d-1\right )-2 a d x^3-b^2+2 b x+d x^4\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(a-x) x^{2/3} \left (x^2-2 b x+a b\right )}{\sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}dx}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(a-x) x^{4/3} \left (x^2-2 b x+a b\right )}{\sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(a-x) x^{4/3} \left (x^2-2 b x+a b\right )}{\sqrt [3]{-\left ((a-x) (x-b)^2\right )} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(a-x)^{2/3} x^{4/3} \left (x^2-2 b x+a b\right )}{(x-b)^{2/3} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a-x)^{2/3} x^{10/3}}{(x-b)^{2/3} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}+\frac {2 b (a-x)^{2/3} x^{7/3}}{(x-b)^{2/3} \left (d x^4-2 a d x^3-\left (1-a^2 d\right ) x^2+2 b x-b^2\right )}+\frac {a b (a-x)^{2/3} x^{4/3}}{(x-b)^{2/3} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (a b \int \frac {(a-x)^{2/3} x^{4/3}}{(x-b)^{2/3} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}d\sqrt [3]{x}+\int \frac {(a-x)^{2/3} x^{10/3}}{(x-b)^{2/3} \left (-d x^4+2 a d x^3+\left (1-a^2 d\right ) x^2-2 b x+b^2\right )}d\sqrt [3]{x}+2 b \int \frac {(a-x)^{2/3} x^{7/3}}{(x-b)^{2/3} \left (d x^4-2 a d x^3-\left (1-a^2 d\right ) x^2+2 b x-b^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

input

Int[(x*(-a + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-b^2 
+ 2*b*x + (-1 + a^2*d)*x^2 - 2*a*d*x^3 + d*x^4)),x]
 

output

$Aborted
 

3.31.15.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.31.15.4 Maple [F]

\[\int \frac {x \left (-a +x \right ) \left (a b -2 b x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-b^{2}+2 b x +\left (a^{2} d -1\right ) x^{2}-2 a d \,x^{3}+d \,x^{4}\right )}d x\]

input

int(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+(a^2*d- 
1)*x^2-2*a*d*x^3+d*x^4),x)
 

output

int(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+(a^2*d- 
1)*x^2-2*a*d*x^3+d*x^4),x)
 

3.31.15.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+( 
a^2*d-1)*x^2-2*a*d*x^3+d*x^4),x, algorithm="fricas")
 

output

Timed out
 

3.31.15.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(-a+x)*(a*b-2*b*x+x**2)/(x*(-a+x)*(-b+x)**2)**(1/3)/(-b**2+2*b 
*x+(a**2*d-1)*x**2-2*a*d*x**3+d*x**4),x)
 

output

Timed out
 

3.31.15.7 Maxima [F]

\[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, b x + x^{2}\right )} {\left (a - x\right )} x}{{\left (2 \, a d x^{3} - d x^{4} - {\left (a^{2} d - 1\right )} x^{2} + b^{2} - 2 \, b x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+( 
a^2*d-1)*x^2-2*a*d*x^3+d*x^4),x, algorithm="maxima")
 

output

integrate((a*b - 2*b*x + x^2)*(a - x)*x/((2*a*d*x^3 - d*x^4 - (a^2*d - 1)* 
x^2 + b^2 - 2*b*x)*(-(a - x)*(b - x)^2*x)^(1/3)), x)
 

3.31.15.8 Giac [F]

\[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, b x + x^{2}\right )} {\left (a - x\right )} x}{{\left (2 \, a d x^{3} - d x^{4} - {\left (a^{2} d - 1\right )} x^{2} + b^{2} - 2 \, b x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(-a+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-b^2+2*b*x+( 
a^2*d-1)*x^2-2*a*d*x^3+d*x^4),x, algorithm="giac")
 

output

integrate((a*b - 2*b*x + x^2)*(a - x)*x/((2*a*d*x^3 - d*x^4 - (a^2*d - 1)* 
x^2 + b^2 - 2*b*x)*(-(a - x)*(b - x)^2*x)^(1/3)), x)
 

3.31.15.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x (-a+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2+2 b x+\left (-1+a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\int -\frac {x\,\left (a-x\right )\,\left (x^2-2\,b\,x+a\,b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (-b^2+2\,b\,x+d\,x^4-2\,a\,d\,x^3+\left (a^2\,d-1\right )\,x^2\right )} \,d x \]

input

int(-(x*(a - x)*(a*b - 2*b*x + x^2))/((-x*(a - x)*(b - x)^2)^(1/3)*(x^2*(a 
^2*d - 1) + 2*b*x + d*x^4 - b^2 - 2*a*d*x^3)),x)
 

output

int(-(x*(a - x)*(a*b - 2*b*x + x^2))/((-x*(a - x)*(b - x)^2)^(1/3)*(x^2*(a 
^2*d - 1) + 2*b*x + d*x^4 - b^2 - 2*a*d*x^3)), x)